Processor-oblivious parallel algorithms

and scheduling

Illustration on parallel prefix Jean-Louis Roch, Daouda Traore

INRIA-CNRS Moais team - LIG Grenoble, France

Contents

I. What is a processor-oblivious parallel algorithm ? II. Work-stealing scheduling of parallel algorithms III. Processor-oblivious parallel prefix computation

Workshop “Scheduling Algorithms for New Emerging Applications” - CIRM Luminy -May 29th-June 2nd, 2006

Dynamic architecture : non-fixed number of resources, variable speedseg: grid, … but not only: SMP server in multi-users mode

The problemProblem: compute f(a)

Sequentialalgorithm

parallelP=2

parallelP=100

parallelP=max

...

Multi-user SMP server GridHeterogeneous network

?Which algorithm to choose ?

… …

Dynamic architecture : non-fixed number of resources, variable speedseg: grid, … but not only: SMP server in multi-users mode

=> motivates « processor-oblivious » parallel algorithm that : + is independent from the underlying architecture:

no reference to p nor i(t) = speed of processor i at time t nor …

+ on a given architecture, has performance guarantees : behaves as well as an optimal (off-line, non-oblivious) one

Problem: often, the larger the parallel degree, the larger the #operations to perform !

Processor-oblivious algorithms

• Prefix problem : • input : a0, a1, …, an • output : 0, 1, …, n with

• Sequential algorithm : for (i= 0 ; i <= n; i++ ) [ i ] = [ i – 1 ] * a [ i ] ;

• Fine grain optimal parallel algorithm [Ladner-Fischer]:

Prefix computation

Critical time W =2. log n

but performs W1 = 2.n ops

Twice more expensive   than the sequential …

a0 a1 a2 a3 a4 … an-1 an

* * **

Prefix of size n/2 1 3 … n

2 4 … n-1

** *

performs W1 = W = n operations

• Any parallel algorithm with critical time W runs on p processors in time

– strict lower bound : block algorithm + pipeline [Nicolau&al. 1996]

–Question : How to design a generic parallel algorithm, independent from the architecture, that achieves optimal performance on any given architecture ? –> to design a malleable algorithm where scheduling suits the number of operations performed to the architecture

Prefix computation : an example where parallelism always costs

- Heterogeneous processors with changing speed [Bender-Rabin02]

=> i(t) = instantaneous speed of processor i at time t in #operations per second

- Average speed per processor for a computation with duration T :

- Lower bound for the time of prefix computation :

Architecture model

Work-stealing (1/2)

«Depth »

W = #ops on a critical path

(parallel time on resources)

• Workstealing = “greedy” schedule but distributed and randomized

• Each processor manages locally the tasks it creates• When idle, a processor steals the oldest ready task on a

remote -non idle- victim processor (randomly chosen)

« Work »

W1= #total

operations performed

Work-stealing (2/2)

«Depth »

W = #ops on a critical path

(parallel time on resources)

« Work »

W1= #total

operations performed

• Interests : -> suited to heterogeneous architectures with slight modification [Bender-Rabin02]

-> if W small enough near-optimal processor-oblivious schedule with good probability on p processors with average speeds ave

NB : #succeeded steals = #task migrations < p W [Blumofe 98, Narlikar 01, Bender 02]

• Implementation: work-first principle [Cilk serie-parallel, Kaapi dataflow] -> Move scheduling overhead on the steal operations (infrequent case)-> General case : “local parallelism” implemented by sequential function call

• General approach: to mix both • a sequential algorithm with optimal work W1 • and a fine grain parallel algorithm with minimal critical time W

• Folk technique : parallel, than sequential • Parallel algorithm until a certain « grain »; then use the sequential one• Drawback : W increases ;o) …and, also, the number of steals

• Work-preserving speed-up technique [Bini-Pan94] sequential, then parallel Cascading [Jaja92] : Careful interplay of both algorithms to build one with both

W small and W1 = O( Wseq ) • Use the work-optimal sequential algorithm to reduce the size • Then use the time-optimal parallel algorithm to decrease the time

Drawback : sequential at coarse grain and parallel at fine grain ;o(

How to get both optimal work W1 and W small?

Alternative : concurrently sequential and parallelBased on the Work-first principle : Executes always a sequential algorithm to reduce parallelism overhead

use parallel algorithm only if a processor becomes idle (ie steals) by extracting parallelism from a sequential computation

Hypothesis : two algorithms : • - 1 sequential : SeqCompute

- 1 parallel : LastPartComputation : at any time, it is possible to extract parallelism from the remaining computations of the sequential algorithm

– Self-adaptive granularity based on work-stealingSeqCompute

Extract_parLastPartComputation

SeqCompute

Parallel

Sequential

0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12

Work-stealer 1

MainSeq.

Work-stealer 2

Adaptive Prefix on 3 processors

1

Steal request

Parallel

Sequential

Adaptive Prefix on 3 processors

0 a1 a2 a3 a4

Work-stealer 1

MainSeq. 1

Work-stealer 2

a5 a6 a7 a8

a9 a10 a11 a127

3

Steal request

2

6 i=a5*…*ai

Parallel

Sequential

Adaptive Prefix on 3 processors

0 a1 a2 a3 a4

Work-stealer 1

MainSeq. 1

Work-stealer 2

a5 a6 a7 a8

7

3 42

6 i=a5*…*ai

a9 a10 a11 a12

8

4

Preempt

10 i=a9*…*ai

8

8

Parallel

Sequential

Adaptive Prefix on 3 processors

0 a1 a2 a3 a4 8

Work-stealer 1

MainSeq. 1

Work-stealer 2

a5 a6 a7 a8

7

3 42

6 i=a5*…*ai

a9 a10 a11 a12

85

10 i=a9*…*ai9

6

11

8

Preempt 11

118

Parallel

Sequential

Adaptive Prefix on 3 processors

0 a1 a2 a3 a4 8 11 a12

Work-stealer 1

MainSeq. 1

Work-stealer 2

a5 a6 a7 a8

7

3 42

6 i=a5*…*ai

a9 a10 a11 a12

85

10 i=a9*…*ai9

6

11

12

10

7

118

Parallel

Sequential

Adaptive Prefix on 3 processors

0 a1 a2 a3 a4 8 11 a12

Work-stealer 1

MainSeq. 1

Work-stealer 2

a5 a6 a7 a8

7

3 42

6 i=a5*…*ai

a9 a10 a11 a12

85

10 i=a9*…*ai9

6

11

12

10

7

118

Implicit critical path on the sequential process

Analysis of the algorithm • Execution time

• Sketch of the proof :– Dynamic coupling of two algorithms that completes simultaneously:– Sequential: (optimal) number of operations S on one processor– Parallel : minimal time but performs X operations on other processors

• dynamic splitting always possible till finest grain BUT local sequential– Critical path small ( eg : log X)– Each non constant time task can potentially be splitted (variable speeds)

– Algorithmic scheme ensures Ts = Tp + O(log X)=> enables to bound the whole number X of operations performedand the overhead of parallelism = (s+X) - #ops_optimal

Lower bound

Adaptive prefix : experiments1

Single-user context : processor-oblivious prefix achieves near-optimal performance : - close to the lower bound both on 1 proc and on p processors

- Less sensitive to system overhead : even better than the theoretically “optimal” off-line parallel algorithm on p processors :

Optimal off-line on p procs

Oblivious

Prefix sum of 8.106 double on a SMP 8 procs (IA64 1.5GHz/ linux)T

ime

(s)

#processors

Pure sequential

Single user context

Adaptive prefix : experiments 2

Multi-user context : Additional external charge: (9-p) additional external dummy processes are concurrently executed Processor-oblivious prefix computation is always the fastest 15% benefit over a parallel algorithm for p processors with off-line schedule,

Multi-user context : Additional external charge: (9-p) additional external dummy processes are concurrently executed Processor-oblivious prefix computation is always the fastest 15% benefit over a parallel algorithm for p processors with off-line schedule,

External charge (9-p external processes)

Off-line parallel algorithm for p processors

Oblivious

Prefix sum of 8.106 double on a SMP 8 procs (IA64 1.5GHz/ linux)

Tim

e (s

)

#processors

Multi-user context :

Conclusion

The interplay of an on-line parallel algorithm directed by work-stealing schedule is useful for the design of processor-oblivious algorithms

Application to prefix computation : - theoretically reaches the lower bound on heterogeneous processors

with changing speeds - practically, achieves near-optimal performances on multi-user SMPs

Generic adaptive scheme to implement parallel algorithms with provable performance

- work in progress : parallel 3D reconstruction [oct-tree scheme with deadline constraint]

Thank you !

QuickTime™ et undécompresseur codec YUV420

sont requis pour visionner cette image.

Interactive Distributed Simulation[B Raffin &E Boyer]

- 5 cameras, - 6 PCs

3D-reconstruction+ simulation+ rendering

->Adaptive scheme to maximize 3D-reconstruction precision within fixed timestamp

The Prefix race: sequential/parallel fixed/ adaptive

Race between 9 algorithms (44 processes) on an octo-SMPSMP

0 5 10 15 20 25

1

2

3

4

5

6

7

8

9

Execution time (seconds)

Série1

Adaptative 8 proc.

Parallel 8 proc.

Parallel 7 proc.

Parallel 6 proc.Parallel 5 proc.

Parallel 4 proc.

Parallel 3 proc.

Parallel 2 proc.

Sequential

On each of the 10 executions, adaptive completes first

Adaptive prefix : some experiments

Single user contextAdaptive is equivalent to:

- sequential on 1 proc - optimal parallel-2 proc. on 2 processors - … - optimal parallel-8 proc. on 8 processors

Multi-user contextAdaptive is the fastest15% benefit over a static grain algorithm

Multi-user contextAdaptive is the fastest15% benefit over a static grain algorithm

External charge

Parallel

Adaptive

Parallel

Adaptive

Prefix of 10000 elements on a SMP 8 procs (IA64 / linux)

#processorsT

ime

(s)

Tim

e (s

)

#processors

With * = double sum ( r[i]=r[i-1] + x[i] )

Single user Processors with variable speeds

Remark for n=4.096.000 doubles :- “pure” sequential : 0,20 s- minimal ”grain” = 100 doubles : 0.26s on 1 proc

and 0.175 on 2 procs (close to lower bound)

Finest “grain” limited to 1 page = 16384 octets = 2048 double